Thermodynamics Problem and Fourier's Law

[mrbranches]

Hello!

I was wondering if anyone could help me with the following hypothetical problem:

I have a structure consisting of approximately 1 ton of copper. I need to calculate the amount of heat required to raise the temperature of an area of the structure to 200C within a timeframe of 20 mins. This requires taking into consideration heat lost to thermal transfer. I understand that copper has a thermal conductivity of 385 W/m K and that the first step will be to calculate the flux rate by applying Fourier's Law. However, I'm at a loss where to go from here.

If anyone can hep me with this problem I'd greatly appreciate it.

 

[lalbatros]

I think you need to give more information on you problem, like a drawing and some idea about the context. For example, this would allow us to see if your problem is closer to a semi-infinite solid problem or to a thin wire structure problem. Look in a book or on the web about "transient heat conduction".

In the case of a semi-infinite heat conduction problem, the heat flux will decrease with time as k DT/(a.t)^0.5, this is so because the heat flux will penetrate into the solid and the temperature gradient at the surface will decrease with time. The conductivity is k, DT is the temperature difference between the heating medium and the bulk solid, a is the heat diffusivity, t is the time. After some time, the main heat transfer resistance is within the solid itself because of the reduced tempertaure gradients: the overall heat transfer coefficient is determined by the state of the heat transfer within the solid.

In contrast, for a thin wire structure, the internal temperature (within the wire) will essentially be uniform and the heat flux will decrease faster, exponentially. The heat conductivity and the diffusivity will play little role (as long as the structure is thin enough) and the heat exchange will be determined by the external heat exchange coefficient. This will depend on the method of heating: radiant, contact, convective, and the empirical formulas to be used are the main topic of heat transfer handbooks. For example, for convective heat with gas, the heat transfer coefficient depends on the speed of the gas along the solid surface.

Michel

PS: Are you maybe dealing with some kind of heat exchanger?

 

[m2003]

Hi there!

Calculating the amount of heat required to raise the temperature of a material within a specific timeframe is an important problem in thermodynamics. In this case, we can use Fourier's Law to determine the heat flux rate, which is the amount of heat transferred per unit area per unit time. This law states that the heat flux rate is proportional to the temperature gradient and the thermal conductivity of the material.

To solve this problem, we can use the following equation:

q = -k * A * (dT/dx)

Where q is the heat flux rate, k is the thermal conductivity of copper (385 W/m K), A is the area of the structure, and dT/dx is the temperature gradient (change in temperature over distance). We can assume that the temperature gradient is constant over the entire structure, as it is being heated uniformly.

Next, we can calculate the total amount of heat required using the equation:

Q = q * A * t

Where Q is the total amount of heat, q is the heat flux rate calculated from Fourier's Law, A is the area of the structure, and t is the time required (20 mins in this case).

I hope this helps you solve your problem! Let me know if you have any further questions. Good luck!